A generalization of some inequalities for the gamma function

Laforgia (1984) obtained some inequalities of the type Γ(k + λ)Γ(k + 1) > 1(k + α) 1−λ or Γ(k + λ)Γ(k + 1) 1(k + α)1−λ according to the values of the positive parameters α and λ, valid for every non-negative real value of k, or at least for k greater than or equal than a k0 depending on α and λ. In this paper a complete analysis of the problem is carried out, in order to establish, for fixed α and λ, which of the two former inequalities holds, and for which values of k.